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Abstract. It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interior-point methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps. Semidefinite linear programming (SDP) is a generalization of LP where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables. There are many applications, e.g., in systems and control theory and combinatorial optimization. However, the Lagrangian dual for SDP can have a duality gap. We discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming. These duals guarantee strong duality, i.e., a zero duality gap and dual attainment. This paper is motivated by the recent paper by Ramana where one of these duals is introduced.

We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.

informs doi 10.1287/moor.1060.0242

This paper presents a unified study of duality properties for the problem of minimizing a linear function over the intersection of an affine space with a convex cone infinite dimension. Existing duality results are carefully surveyed and some new duality properties are established. Examples are given to illustrate these new properties. The topics covered in this paper include Gordon-Stiemke type theorems, Farkas type theorems, perfect duality, Slater condition, regularization, Ramana's duality, and approximate dualities. The dual representations of various convex sets, convex cones and conic convex programs are also discussed.

Several important problems in control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to Linear Matrix Inequality (LMI) constraints. From convex optimization duality theory, conditions for infeasibility of the LMIs as well as dual optimization problems can be formulated. These can in turn be reinterpreted in control or system theoretic terms, often yielding new results or new proofs for existing results from control theory. We explore such connections for a few problems associated with linear time-invariant systems. 1

The Quadratic Assignment Problem, QAP, is arguably the hardest of the NP-hard problems. One of the main reasons is that it is very difficult to get good quality bounds for branch and bound algorithms. We show that many of the bounds that have appeared in the literature can be ranked and put into a unified Semidefinite Programming, SDP, framework. This is done using redundant quadratic constraints and Lagrangian relaxation. Thus, the final SDP relaxation ends up being the strongest.

The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primal-dual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, and strict complementarity, for linear cone optimization problems in finite dimensions. These results include: weakest and universal constraint qualifications, CQs; duality and characterizations of optimality that hold without any CQ; geometry of nice and devious cones; the geometric relationships between zero duality gaps, strict complementarity, and the facial structure of cones; and, the connection between theory and empirical evidence for lack of a CQand failure of strict complementarity. One theme is the notion of minimal representation of the cone and the constraints in order to regularize the problem and avoid both the theoretical and numerical difficulties that arise due to (near) loss of a CQ. We include a discussion on obtaining these representations efficiently.

For iterative sequences that converge to the solution set of a linear matrix inequality, we show that the distance of the iterates to the solution set is at most O(ffl 2 \Gammad ). The nonnegative integer d is the so--called degree of singularity of the linear matrix inequality, and ffl denotes the amount of constraint violation in the iterate. For infeasible linear matrix inequalities, we show that the minimal norm of ffl--approximate primal solutions is at least 1=O(ffl 1=(2 d \Gamma1) ), and the minimal norm of ffl--approximate Farkas-- type dual solutions is at most O(1=ffl 2 d \Gamma1 ). As an application of these error bounds, we show that for any bounded sequence of ffl--approximate solutions to a semi-definite programming problem, the distance to the optimal solution set is at most O(ffl 2 \Gammak ), where k is the degree of singularity of the optimal solution set. Keywords: semi-definite programming, error bounds, linear matrix inequality, regularized duality. AMS s...

Given the conic formulation of a convex program, we describe a theory that ffl Characterizes the faces of the feasible sets. ffl Defines nondegeneracy, strict complementarity and relates these to the optimal face, analogously to the LP case. ffl Characterizes the tangent spaces of the feasible sets. ffl Introduces the family of boundary structure inequalities which relate the dimensions of the above-mentioned sets. ffl Using the general framework, gives a simple derivation for a number of structural results about problems that can be formulated as an SDP. ffl Shows how two algorithmic aspects can be handled: converting a feasible solution into one, which is also an extreme point; and performing a restricted sensitivity analysis. 1 Introduction Consider the primal-dual pair of optimization problems Min hc; xi Max hb; yi (P ) s:t: x 2 K s:t: z 2 K (D) Ax = b A y + z = c where ffl X and Y are euclidean spaces with dimX dimY . ffl A : X ! Y is a linear operator, assumed to ...